Question

Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10⁸ m. Show that the mass of Jupiter is about one-thousandth that of the sun.

Answer 1

Given: The orbital period of the satellite is 1.769 days and the radius of the orbit is 4.22× 10 8  m.

The mass of the Jupiter is given as,

M 1 = 4 π 2 r 1 3 G T 1 2

Where, the mass of the Jupiter is M 1 , the radius of the orbit of the satellite is r 1 and orbital period of the satellite is T 1 .

By substituting the given values in the above equation, we get

M 1 = 4 π 2 × ( 4.22× 10 8 ) 3 G× ( 1.769×24×60×60 ) 2 (1)

Let the orbital period for earth is 1 year, the radius of the orbit of the earth is 1.496× 10 11  mand mass of earth is M.

The mass of the sun is given as,

M= 4 π 2 r 3 G T 2

By substituting the values we obtain,

M= 4 π 2 × ( 1.496× 10 11 ) 3 G× ( 365.25×24×60×60 ) 2 (2)

On dividing equation (2) by (1), we obtain

M M 1 = 4 π 2 × ( 1.496× 10 11 ) 3 G× ( 365.25×24×60×60 ) 2 × G× ( 1.769×24×60×60 ) 2 4 π 2 × ( 4.22× 10 8 ) 3 =1046 M 1 M = 1 1046 ≃ 1 1000

Thus, the mass of Jupiter is about 1 1000 th that of the Sun.